Theta cycles of modular forms modulo p2

Journal article, 2026

The theta cycle of a modular form modulo a prime p >= 5 is well understood. By contrast, the theta cycle modulo a power of p is still mysterious and experimentally erratic. Here, we completely determine the theta cycle of a weight k[ (p-k+1)/2 & RightFloor; further low points at regular positions. Moreover, we detect low points at exceptional positions which solve a quadratic equation modulo p, and which disturb the otherwise regular structure in the segments that we exhibit.